Exploring Arc Length and Curvature in Calculus

1. 13.3 • Arc Length and Curvature

2. Arc Length: • As studied in Calc 2: • Leibniz notation

3. Arc Length: plane curve (2D) • If we parametrize y = f(x) as follows: • x = t, y = f(t), a tb, then the arc length is: • L = = • A more general parametrization would be: • x = f (t), y = g(t), a tb, then the arc length is:

4. Arc Length: space curve (3D) • Similarly: The length of a space curve (in 3D) is defined as:

5. Arc Length • Notice that both of the arc length formulas, plane or space, can be put into the more compact form: • because, for plane curves r(t) = f (t) i + g(t) j, • and for space curves r(t) = f (t) i + g(t) j + h(t) k,

6. Example 1 • Find the length of the arc of the circular helix with vector equation • r(t) = cos t i + sin t j + t k from the point (1, 0, 0) to the point (1, 0, 2). • Solution: • Since r'(t) = –sin t i + cos t j + k, we have • The arc from (1, 0, 0) to (1, 0, 2) is described by the parameter interval 0t2 and so we have by the formula in the last slide:

7. Arc Length: independent of parametrization! • A single curve C can be represented by more than one parametric vector function. • For example: We can represent the twisted cubic by: • r1(t) = t, t2, t3 1t2 • But we could also represent it by: • r2(u) = eu, e2u, e3u 0u ln 2 • The connection between the parameters t and u is given by t = eu. • We say that Equations 4 and 5 are parametrizations of the curve C. • If we were to compute the length of C using Equations 4 and 5, we would get the same answer!

8. Arc Length • CONCLUSION: • When using the formula: • to compute arc length, the answer is independent of the parametrization that is used.

9. Arc Length Function: • Given that C is a curve with vector function: • r(t) = f (t) i + g(t) j + h(t) ka tb • where r' is continuous and C is traversed exactly once as t increases from a to b, then we define: • We define its arc length function sby • where s(t) is the length of the part of C between r(a) and r(t).

10. Arc Length • If we differentiate both sides using Part 1 of the Fundamental Theorem of Calculus, we obtain • It is often useful to parametrize a curve with respect to arc length s, rather than just a random parameter t, because arc length arises naturally from the shape of the curve and does not depend on a particular coordinate system. Calculations are consequently more convenient.

11. Parametrization with respect to Arc Length • If a curve r(t) is already given in terms of a parameter t and s(t) is the arc length function, then we may be able to solve for t as a function of s: • t = t(s) • Then the curve can be reparametrized in terms of s by substituting for t: • r = r(t(s)). • For example, if s = 3, r(t(3)) is the position vector of the point 3 units of length along the curve from its starting point. • This will be very handy in Chapter 16, when we will talk about Line Integrals and Surface Integrals (which have manyapplications in Physics and Science.)

12. Curvature • A parametrization r(t) is called smooth on an interval I if r' is continuous and r'(t) 0 on I. • A curve is called smooth if it has a smooth parametrization. A smooth curve has no sharp corners or cusps; when the tangent vector turns, it does so continuously. • If C is a smooth curve defined by the vector function r, recall that the unit tangent vector T(t) is given by: • and indicates the direction of the curve.

13. Curvature • The figure below shows that T(t) changes direction very slowly when C is fairly straight, but it changes direction more quickly when C bends or twists more sharply. • Unit tangent vectors at equally • spaced points on C • The curvature of C at a given point is a measure of how quickly the curve changes direction at that point.

14. Curvature • Specifically, we define curvature to be the magnitude of the rate of change of the unit tangent vector with respect to arc length. • (We use arc length so that the curvature will be independent of the parametrization.) • Recall:

15. Curvatures • The curvature is easier to compute if it is expressed in terms of the parameter t instead of s, so we use the ChainRule to write • But ds/dt = | r'(t) | , so:

16. Example 2 • Show that the curvature of a circle of radius a is 1/a. • Solution: • We can take the circle to have center the origin, and then a parametrization is: • r(t) = a cos t i + a sin t j • Therefore r'(t) = –a sin t i + a cos t j and | r'(t) | = a • So then: • And: T'(t) = –cos t i – sin t j

17. Example 2 – Solution • cont’d • This gives | T'(t)| = 1 • Finally, using Equation 9, we have:

18. Curvature • The result of the previous example shows that small circles have large curvature and large circles have small curvature, in accordance with our intuition. • We can see directly from the definition of curvature that the curvature of a straight line is always 0 because the tangent vector is constant. • Although Formula 9 can be used in all cases to compute the curvature, the formula given by theorem 10 is often more convenient to apply.

19. Curvature: formula for a plane curve y=f(x) • For the special case of a plane curve with equation y = f (x), • choose x as the parameter and write: • r(x) = xi + f (x) j. • Then r'(x) = i + f'(x) j and r''(x) = f''(x) j. • Since i  j = k and j  j = 0, it follows that: • r'(x)  r''(x) = f''(x) k. • We also have: and so, by Theorem 10,

20. The Normal and Binormal Vectors • At a given point on a smooth space curve r(t), there are many vectors that are orthogonal to the unit tangent vector T(t). • We define the principal unit normal vector N(t)(or simply unit normal) as:

21. The Normal and Binormal Vectors • The vector B(t) = T(t) N(t) is called the binormal vector. • It is perpendicular to both T and N and is also a unit vector.

22. Example 3 • Find the unit normal and binormal vectors for the circular helix • r(t) = cos t i + sin t j + tk • Solution: • We first compute the ingredients needed for the unit normal vector: • r'(t) = –sin t i + cos t j + k so:

23. Example 3 – Solution • cont’d • The normal vector is: • This shows that the normal vector at a point on the helix is horizontal and points toward the z-axis. • The binormal vector is: • B(t) = T(t) N(t)

24. The Normal and Binormal Vectors • The plane determined by the normal and binormal vectors N and B at a point P on a curve C is called the normal plane of C at P. • It consists of all lines that are orthogonal to the tangent vector T. • The plane determined by the vectors T and N is called the osculating plane of C at P. • (It is the plane that comes closest to containing the part of the curve near P. (For a plane curve, the osculating plane is simply the plane that contains the curve.)

25. The Normal and Binormal Vectors • The circle that lies in the osculating plane of C at P, has the same tangent as C at P, lies on the concave side of C (toward which N points), and has radius  = 1/ (the reciprocal of the curvature) is called the osculating circle (or the circle of curvature) of C at P. • It is the circle that best describes how C behaves near P; it shares the same tangent, normal, and curvature at P.

26. 13.3 Summary of formulas: • Summary of the formulas for unit tangent, unit normal vector, unit binormal vector, and curvature: • Osculating circle radius:  = 1/ • Curvature for a plane curve y=f(x):